fft3d(3F)

NAME
     cfft3d, zfft3d - 3D Complex-to-Complex Fast Fourier Transform.

SYNOPSYS
     Fortran :
     subroutine cfft3d( sign, n1, n2, n3, array, la1, la2, coef )
          integer  sign, n1, n2, n3, la1, la2
          complex  array(la1,la2,n3)
          complex coef((n1+15)+(n2+15)+(n3+15))

     subroutine zfft3d( sign, n1, n2, n3, array, la1, la2, coef )
          integer  sign, n1, n2, n3, la1, la2
          double complex array(la1,la2,n3)
          double complex coef((n1+15)+(n2+15)+(n3+15))


     C :
     #include <fft.h>
     int cfft3d(int sign,int n1,int n2,int n3,complex *array,
                    int la1, int la2, complex *coef);
     int zfft3d(int sign,int n1,int n2,int n3,zomplex *array,
                    int la1, int la2, zomplex *coef);


DESCRIPTION
     cfft3d and zfft3d compute in place the complex Fourier transform of
     complex 3D sequence of size N1xN2xN3.  The value F{j1,j2,j3} of the
     transform of the 3D sequence f{i1,i2,i3} is equal to:
       F{j1,j2,j3} = Sum(W1^(i1*j1)*W2^(i2*j2)*W3^(i3*j3*f{i1,i2,i3} ),
                     for i[123] =0,...,(N[123]-1)
                 W[123] = exp( (Sign*2*sqrt(-1)*PI) / N[123] )


Storage
     It is assumed that the (N1 x N2 x N3) 3D sequence is stored along
     dimension N1.  So the index {i+1,j,l} has an offset of 1 element with
     respect to {i,j,l}, and {i,j+1,k} an offset of la1 elements with respect
     to {i,j,k}, and {i,j,k+1} an offset of (la1 * la2) elements with respect
     to {i,j,k}.


Algorithm
     The complex-to-complex 3D Fourier transform is computed with a row-column
     approach.
      - First, N3 2D FFTs complex-to-complex of size N1xN2 are evaluated,
      - then, N1*N2 FFTs complex-to-complex of size N3 are performed,
     stride=la1*la2, and leading_dimension=1.

PARAMETERS
     SIGN - Integer specifying which sign to be used for the expression of W
     (see above) - must be either +1 or -1.
     Unchanged on exit.

     N1 Integer, the first dimension size of the 3D sequence.  Unchanged on
     exit.

     N2 Integer, the second dimension size of the 3D sequence.  Unchanged on
     exit.

     N3 Integer, the third dimension size of the 3D sequence.  Unchanged on
     exit.

     ARRAY Array containing the samples of the 3D sequence to be transformed.
     On input, the element {i,j,k} of the sequence is stored as A(i,j,k) in
     Fortran , and A[i+j*la1+k*la1*la2] in C. On exit, the array is
     overwritten.

     LA1 Integer, first leading dimension: increment between the samples of
     two consecutive 1D sub-sequences (e.g between {i,j+1,k} and {i,j,k} ).
     Unchanged on exit.

     LA2 Integer, second leading dimension: number of the 1D sub-sequence
     between two consecutive 2D sub-sequences (e.g between {i,j,k+1} and
     {i,j,k}).  Unchanged on exit.

     COEFF - Array of at least ( (N+15)+(N2+15)+(N3+15) ) elements.  On entry
     it contains the Sines/Cosines and factorization of N. COEFF needs to be
     initialized with a call to cfft3di or zfft3di. Unchanged on exit.


Example of Calling Sequence
     3D FFT computed on a complex sequence of size 100x64x125.
     Fortran
          complex array(0:100-1,0:64-1,0:125-1)
          complex coeff(100+15 + 64+15 + 125+15)
          call cfft3di( 100, 64, 125, coeff)
          call cfft3d( -1, 100, 64, 125, array, 100, 64, coeff)

     C
          #include <fft.h>
          complex array[100*64*125], *coeff;
          coeff = cfft3di( 100, 64, 125, NULL);
          cfft3d( -1, 100, 64, 125, array, 100, 64, coeff);

SEE ALSO
     fft, cfft3di, zfft3di